Precalculus Examples

 $\begin{matrix} Side & Angle \\ \begin{matrix} b = & 14 \\ c = & 17 \\ a = & 11 \end{matrix} & \begin{matrix} A = \\ B = \\ C = \end{matrix} \end{matrix}$

Step 1

Use the law of cosines to find the unknown side of the triangle, given the other two sides and the included angle.

$a^{2} = b^{2} + c^{2} - 2 b c \cos (A)$

Step 2

Solve the equation.

$A = \arccos (\frac{b^{2} + c^{2} - a^{2}}{2 b c})$

Step 3

Substitute the known values into the equation.

$A = \arccos (\frac{{(14)}^{2} + {(17)}^{2} - {(11)}^{2}}{2 (14) (17)})$

Step 4

Simplify the results.

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Step 4.1

Simplify the numerator.

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Step 4.1.1

Raise $14$ to the power of $2$ .

$A = \arccos (\frac{196 + 17^{2} - 11^{2}}{2 (14) \cdot 17})$

Step 4.1.2

Raise $17$ to the power of $2$ .

$A = \arccos (\frac{196 + 289 - 11^{2}}{2 (14) \cdot 17})$

Step 4.1.3

Raise $11$ to the power of $2$ .

$A = \arccos (\frac{196 + 289 - 1 \cdot 121}{2 (14) \cdot 17})$

Step 4.1.4

Multiply $- 1$ by $121$ .

$A = \arccos (\frac{196 + 289 - 121}{2 (14) \cdot 17})$

Step 4.1.5

Add $196$ and $289$ .

$A = \arccos (\frac{485 - 121}{2 (14) \cdot 17})$

Step 4.1.6

Subtract $121$ from $485$ .

$A = \arccos (\frac{364}{2 (14) \cdot 17})$

Step 4.2

Simplify the denominator.

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Step 4.2.1

Multiply $2$ by $14$ .

$A = \arccos (\frac{364}{28 \cdot 17})$

Step 4.2.2

Multiply $28$ by $17$ .

$A = \arccos (\frac{364}{476})$

Step 4.3

Cancel the common factor of $364$ and $476$ .

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Step 4.3.1

Factor $28$ out of $364$ .

$A = \arccos (\frac{28 (13)}{476})$

Step 4.3.2

Cancel the common factors.

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Step 4.3.2.1

Factor $28$ out of $476$ .

$A = \arccos (\frac{28 \cdot 13}{28 \cdot 17})$

Step 4.3.2.2

Cancel the common factor.

$A = \arccos (\frac{28 \cdot 13}{28 \cdot 17})$

Step 4.3.2.3

Rewrite the expression.

$A = \arccos (\frac{13}{17})$

Step 4.4

Evaluate $\arccos (\frac{13}{17})$ .

$A = 40.11916689$

Step 5

Use the law of cosines to find the unknown side of the triangle, given the other two sides and the included angle.

$b^{2} = a^{2} + c^{2} - 2 a c \cos (B)$

Step 6

Solve the equation.

$B = \arccos (\frac{a^{2} + c^{2} - b^{2}}{2 a c})$

Step 7

Substitute the known values into the equation.

$B = \arccos (\frac{{(11)}^{2} + {(17)}^{2} - {(14)}^{2}}{2 (11) (17)})$

Step 8

Simplify the results.

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Step 8.1

Simplify the numerator.

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Step 8.1.1

Raise $11$ to the power of $2$ .

$B = \arccos (\frac{121 + 17^{2} - 14^{2}}{2 (11) \cdot 17})$

Step 8.1.2

Raise $17$ to the power of $2$ .

$B = \arccos (\frac{121 + 289 - 14^{2}}{2 (11) \cdot 17})$

Step 8.1.3

Raise $14$ to the power of $2$ .

$B = \arccos (\frac{121 + 289 - 1 \cdot 196}{2 (11) \cdot 17})$

Step 8.1.4

Multiply $- 1$ by $196$ .

$B = \arccos (\frac{121 + 289 - 196}{2 (11) \cdot 17})$

Step 8.1.5

Add $121$ and $289$ .

$B = \arccos (\frac{410 - 196}{2 (11) \cdot 17})$

Step 8.1.6

Subtract $196$ from $410$ .

$B = \arccos (\frac{214}{2 (11) \cdot 17})$

Step 8.2

Simplify the denominator.

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Step 8.2.1

Multiply $2$ by $11$ .

$B = \arccos (\frac{214}{22 \cdot 17})$

Step 8.2.2

Multiply $22$ by $17$ .

$B = \arccos (\frac{214}{374})$

Step 8.3

Cancel the common factor of $214$ and $374$ .

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Step 8.3.1

Factor $2$ out of $214$ .

$B = \arccos (\frac{2 (107)}{374})$

Step 8.3.2

Cancel the common factors.

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Step 8.3.2.1

Factor $2$ out of $374$ .

$B = \arccos (\frac{2 \cdot 107}{2 \cdot 187})$

Step 8.3.2.2

Cancel the common factor.

$B = \arccos (\frac{2 \cdot 107}{2 \cdot 187})$

Step 8.3.2.3

Rewrite the expression.

$B = \arccos (\frac{107}{187})$

Step 8.4

Evaluate $\arccos (\frac{107}{187})$ .

$B = 55.09674167$

Step 9

The sum of all the angles in a triangle is $180$ degrees.

$40.11916689 + C + 55.09674167 = 180$

Step 10

Solve the equation for $C$ .

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Step 10.1

Add $40.11916689$ and $55.09674167$ .

$C + 95.21590857 = 180$

Step 10.2

Move all terms not containing $C$ to the right side of the equation.

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Step 10.2.1

Subtract $95.21590857$ from both sides of the equation.

$C = 180 - 95.21590857$

Step 10.2.2

Subtract $95.21590857$ from $180$ .

$C = 84.78409142$

Step 11

These are the results for all angles and sides for the given triangle.

$A = 40.11916689$

$B = 55.09674167$

$C = 84.78409142$

$a = 11$

$b = 14$

$c = 17$